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**Additional info for Aspects of positivity in functional analysis: proceedings of the conference held on the occasion of H.H. Schaefer's 60th birthday, Tubingen, 24-28 June 1985**

**Example text**

3. 1, if M , N are strongly separating function subspaces, then the map p in (28) is a homeomorphism from d N onto dM. This is very nice result and will be of use in Chapter 6. Everything that we have been discussing in the last few paragraphs is in the paper of Araujo and Font [lo], although we have changed the notation somewhat. The authors' arguments are not based on the usual concepts of T-sets or extreme points of the dual balls, but are closer to the approach of Stone. The proofs are lengthy and we have chosen to omit them.

Suppose p ( c h ( N ) ) is dense in Q and T f = 0 for some f E C ( Q ) . Let s E Q , and suppose t > 0 is given. By continuity of f there is a neighborhood U of s so that I f ( s ) - f ( s l ) l < t for s' E U . ) f ( ~ ( t ) )=l I f ( s ) - h ( t ) ~ f ( t ) l = I f ( s ) l . We conclude that f ( s ) = 0 so that T is one-to-one, and so is an isometry. t - O 2003 by Chapman & HallICRC 42 2. CONTINUOUS FUNCTION SPACES-THE BANACH-STONE THEOREM On the other hand if p ( c h ( N ) ) is not dense, we can find a nonzero f E C ( Q ) which is zero on the closure of p ( c h ( N ) ) so that T f = 0 and T is not injective.

Is compact. 4411 is well known and avoids measure theoretic arguments. : is compact and E a Banach space. Lau [194] quotes the result in this case and attributes it to Lazar [195]. Brosowski and Deutsch [47] give credit to P. D. Morris for helping in their proof. Behrends [23] gives a proof in the context of Banach modules; Tate [302] gives a characterization for certain commutative algebras over R. 1451. 17(i) was given by Singer to show that an extension of a functional might be an extreme point even though the original is not.